Weighted difference of g-factors of light Li-like and H-like ions for an improved determination
of the fine-structure constant
V. A. Yerokhin,1, 2 E. Berseneva,1, 3 Z. Harman,1 I. I. Tupitsyn,3 and C. H. Keitel1
1
arXiv:1606.08620v1 [physics.atom-ph] 28 Jun 2016
2
Max Planck Institute for Nuclear Physics, Saupfercheckweg 1, 69117 Heidelberg, Germany
Center for Advanced Studies, Peter the Great St. Petersburg Polytechnic University, 195251 St. Petersburg, Russia
3
Department of Physics, St. Petersburg State University,
7/9 Universitetskaya naberezhnaya, St. Petersburg 199034, Russia
A weighted difference of the g-factors of the Li- and H-like ion of the same element is studied and optimized
in order to maximize the cancellation of nuclear effects. To this end, a detailed theoretical investigation is
performed for the finite nuclear size correction to the one-electron g-factor, the one- and two-photon exchange
effects, and the QED effects. The coefficients of the Zα expansion of these corrections are determined, which
allows us to set up the optimal definition of the weighted difference. It is demonstrated that, for moderately
light elements, such weighted difference is nearly free from uncertainties associated with nuclear effects and
can be utilized to extract the fine-structure constant from bound-electron g-factor experiments with an accuracy
competitive with or better than its current literature value.
I.
INTRODUCTION
Modern measurements of the bound-electron g-factor in
H-like ions have reached the level of fractional accuracy of
3 × 10−11 [1]. Experiments have also been performed with
Li-like ions [2]. In future it shall be possible to conduct similar experiments not only with a single ion in the trap, but also
with several ions simultaneously. Such a setup would allow
one to directly access differences of the g-factors of different
ions, thus largely reducing systematic uncertainties and possibly gaining about two orders of magnitude in experimental
accuracy [3]. So, experimental investigations of differences of
the bound-electron g factors on a sub-10−12 level look feasible in the future. Such measurements would become sensitive
to the uncertainty of the fine-structure constant α, which is
presently known up to the fractional accuracy of 3 × 10−10
[4]. It might be tempting to use such future experiments as a
tool for an independent determination of α.
In order to accomplish a competitive determination of α
from the bound-electron g-factor experiments, one has to
complete theoretical calculations to a matching accuracy,
which is a challenging task. One of the important problems on
the way is the uncertainty due to nuclear effects, which cannot
be well understood at present. These uncertainties set a limitation on the ultimate accuracy of the theoretical description
and, therefore, on the determination of α.
There is a way to reduce the nuclear effects and the associated uncertainties, by forming differences of different charge
states of the same element. In Ref. [5], it was suggested to use
a weighted difference of the g-factors of the H- and Li-like
ions of the same element in order to suppress the nuclear size
effects by about two orders of magnitude for high-Z ions. In
Ref. [6], a weighted difference of the g-factors of B-like and
H-like charge states of the same element was proposed. It was
shown that the theoretical uncertainty of the nuclear size effect for ions around Pb can be reduced to 4 × 10−10 , which
was several times smaller than the uncertainty due to the finestructure constant at the time of publication of Ref. [6]. Since
then, however, the uncertainty of α was decreased by an order
of magnitude [7–9], thus making it more difficult to access it
in the bound-electron g-factor experiments. In our recent Letter [10] we proposed a weighted difference of the g-factors of
low-Z Li-like and H-like ions, for which a more significant
cancellation of nuclear effects can be achieved. In the present
paper we describe details of the underlying calculations and
report extended numerical results for the finite nuclear size
corrections.
In our approach, the weight Ξ of the specific difference of
the g-factors is determined on the basis of studying the Zα
and 1/Z expansions of various finite nuclear size (fns) corrections, in such a way that the cancellation of these undesirable contributions is maximized. We introduce the following
Ξ-weighted difference of the bound-electron g-factors of the
Li-like and H-like charge states of the same element,
δΞ g = g(2s) − Ξ g(1s) ,
(1)
where g(2s) is the g-factor of the Li-like ion, g(1s) is the gfactor of the H-like ion, and the parameter Ξ is defined as
Ξ=2
−2γ−1
3
1 + (Zα)2
16
2851 1
107 1
1−
+
1000 Z
100 Z 2
,
(2)
p
with the notation γ = 1 − (Zα)2 . The justification of this
choice of Ξ will be given later, after studying the contributions
of individual physical terms to the fns effect.
This article is organized as follows. In Section II we describe our calculations of various fns contributions, namely,
the leading one-electron fns effect, the fns correction from the
one-electron QED effects, and the two- and three-electron fns
corrections due to the exchange of one or more photons between the electrons. The resulting weighted difference of the
g-factors and its utility in determining the fine-structure constant are discussed in Section III, which is followed by a short
conclusion.
2
II.
FINITE NUCLEAR SIZE CORRECTIONS
A.
One-electron finite nuclear size
The leading one-electron fns correction to the boundelectron g-factor is defined as follows:
(0)
(0)
(0)
δgN = gext − gpnt ,
(0)
gext
(3)
(0)
gpnt
where
and
are the leading-order bound-electron g
factor values calculated assuming the extended and the pointlike nuclear models, respectively. The leading-order boundelectron g factor is obtained for ns states as
Z
8 ∞
g (0) = −
dr r3 ga (r) fa (r) ,
(4)
3 0
where ga and fa are the upper and the lower radial components of the ns Dirac wave function, respectively [11].
(0)
The fns correction δgN has an approximate relation to the
corresponding correction to the Dirac energy, which reads
[12] for ns states as
(0)
δgN =
4
δEN
(2γ + 1)
,
3
m
(5)
where δEN is the nuclear-size correction to the Dirac energy.
Eq. (5) is exact in the nonrelativistic limit and also holds with
a reasonable accuracy in the whole region of nuclear charge
numbers Z. Using Eq. (5) and the result of Ref. [13] for δEN ,
the leading one-electron fns effect for ns states can be parameterized as
2γ
i
(Zα)2 h
2 2 Zα Rsph
(0)
1 + (Zα)2 Hn(0,2+) ,
δgN =
5
n
n
(6)
p
where Rsph = 5/3 R is the radius of the nuclear sphere
(0,2+)
with the root-mean-square (rms) charge radius R and Hn
is the remainder due to relativistic effects. The superscript
(0, 2+) indicates that its contribution is of zeroth order in 1/Z
and of second and higher orders in Zα. The nonrelativistic
limit of Eq. (6) agrees with the well-known result of Refs.
[14, 15].
(0,2)
The leading relativistic correction Hn
has been given in
a closed analytical form in Ref. [15]. We deduce from it that
the difference of the relativistic corrections of relative order
(Zα)2 for 2s and 1s states does not depend on the nuclear
charge radius nor on the nuclear charge distribution model,
and is just a constant:
(0,2)
H21
(0,2)
≡ H2
(0,2)
− H1
=
3
.
16
(7)
In the present work we calculate the nuclear-size correc(0)
tion δgN numerically. For the extended nucleus, the radial
Dirac equation is solved with the Dual Kinetic Balance (DKB)
(0)
method [16], which allows us to determine gext with a very
high accuracy. The nuclear-size correction is obtained by subtracting the analytical point-nucleus result. In order to avoid
loss of numerical accuracy in the low-Z region, we used the
DKB method implemented in the quadruple (about 32 digits)
arithmetics.
In our calculations, we used three models of the nuclear
charge distribution. The two-parameter Fermi model is given
by
ρFer (r) =
N
,
1 + exp[(r − r0 )/a]
(8)
where r0 and a are the parameters of the Fermi distribution,
and N is the normalization factor. The parameter a was
fixed by the standard choice of a = 2.3/(4 ln 3) ≈ 0.52 fm.
The homogeneously charged sphere distribution of the nuclear
charge is given by
ρSph (r) =
3
θ(RSph − r) ,
3
4πRsph
(9)
where θ is the Heaviside step function. The Gauss distribution
of the nuclear charge reads
3/2
3 r2
3
(10)
exp − 2 .
ρGauss (r) =
2πR2
2R
The results of our calculations for the 2s and 1s states
are presented in Table I, expressed in terms of the function
(0,2+)
. Experimental values of the rms nuclear charge radii
Hn
R are taken from Ref. [17]. For ions with Z ≥ 10, we perform
calculations with the Fermi and the homogeneously charged
sphere models. The difference of the values obtained with
these two models is taken as an estimation of the model dependence of the results. For ions with Z < 10, the Fermi
model is no longer adequate and we use the Gauss model instead.
We observe that the model dependence of the relativistic
(0,2+)
is generally not negligible; it varies
fns correction Hn
from 1% in the medium-Z region to 5% in the low-Z region.
(0,2+)
However, the model dependence of the difference H2
−
(0,2+)
H1
is tiny. According to Eq. (7), it is suppressed by a
small factor of (Zα)2 . Our calculations show that in addition
it is suppressed by a small numerical coefficient.
We conclude that both the model dependence and the R uncertainty of the one-electron fns correction can be cancelled
up to a very high accuracy by forming a suitably chosen difference. The following weighted difference of the 2s and 1s
one-electron g-factors cancels the one-electron fns contributions of relative orders (Zα)0 and (Zα)2 ,
δΞ0 g = g (0) (2s) − Ξ0 g (0) (1s) ,
(11)
with the weight
Ξ0 = 2
−2γ−1
3
2
1 + (Zα) .
16
(12)
The one-electron fns effects in the difference δΞ0 g arise only
in the relative order (Zα)4 , with a numerically small coefficient.
3
(0,2+)
defined by Eq. (6), for the 2s state (n = 2) and the 1s state (n = 1), for
TABLE I: The relativistic fns correction, in terms of function Hn
different models of the nuclear charge distribution. The rms charge radii R and their errors are taken from the compilation of Ref. [17].
Z
6
R [fm]
2.4702(22)
8
2.6991(52)
10
3.0055(21)
12
3.0570(16)
14
3.1224(24)
20
3.4776(19)
25
3.7057(22)
30
3.9283(15)
35
4.1629(21)
40
4.2694(10)
45
4.4945(23)
50
4.6519(21)
55
4.8041(46)
60
4.9123(25)
(0,2+)
H2
0.9296(3)
0.9827(3)
0.9912(6)
1.0408(5)
1.0248
1.0700(2)
1.0690
1.1067(1)
1.1001(1)
1.1327(1)
1.1542(1)
1.1764(1)
1.1843
1.2030(1)
1.2085
1.2246
1.2297(1)
1.2438
1.2518
1.2652(1)
1.2714(1)
1.2834(1)
1.2920
1.3033(1)
1.3129(1)
1.3235(1)
1.3346(1)
1.3447
Model
Gauss
Sphere
Gauss
Sphere
Fermi
Sphere
Fermi
Sphere
Fermi
Sphere
Fermi
Sphere
Fermi
Sphere
Fermi
Sphere
Fermi
Sphere
Fermi
Sphere
Fermi
Sphere
Fermi
Sphere
Fermi
Sphere
Fermi
Sphere
B. One-electron QED fns correction
(0)
The one-electron QED fns correction δgNQED to the boundelectron g factor can be conveniently parameterized by means
(0)
of the dimensionless function GNQED [18],
(0)
(0)
δgNQED = δgN
α (0)
(Zα, R) ,
G
π NQED
(13)
(0,2+)
(0,2+)
H1
0.7421(3)
0.7951(3)
0.8035(5)
0.8531(5)
0.8370
0.8822(2)
0.8810
0.9186(1)
0.9118(1)
0.9443(1)
0.9647(1)
0.9868(1)
0.9934
1.0119(1)
1.0159
1.0319(1)
1.0350
1.0490(1)
1.0548
1.0679
1.0718
1.0836(1)
1.0897
1.1006
1.1077
1.1180(1)
1.1265
1.1363(1)
H2
(0,2+)
− H1
− 3/16
0.00003
0.00007
0.0001
0.0002
0.0003
0.0003
0.0005
0.0006
0.0008
0.0009
0.0020
0.0021
0.0034
0.0035
0.0051
0.0053
0.0071
0.0073
0.0095
0.0098
0.0121
0.0123
0.0148
0.0151
0.0177
0.0180
0.0206
0.0209
for the 1s state of H-like ions. In the present work, we extend
our calculations to the 2s state, which is required for describing the Li-like ions. The numerical results obtained for the
2s state are listed in Table II. The results for the 1s state are
taken from Ref. [18]. We observe that the QED fns corrections
for the 1s and 2s states, expressed in terms of the function
(0)
GNQED , are very close to each other. Therefore, they largely
cancel in the weighted difference δΞ0 g introduced in Eq. (11).
(0)
where δgN is the leading-order fns correction discussed in
(0)
Sec. II A, and GNQED is a slowly varying function. The correction can be divided into four parts,
(0)
GNQED = GNSE + GNUe,el + GNWK,el + GNVP,ml , (14)
where GNSE is the contribution of the electron self-energy,
GNUe,el is induced by the insertion of the Uehling potential
into the electron line, GNWK,el is the analogous correction by
the Wichmann-Kroll potential, and GNVP,ml is the so-called
magnetic-loop vacuum-polarization correction.
The QED fns correction was studied in detail in our previous investigation [18], where we reported numerical results
C. One-photon exchange fns correction
The one-photon exchange fns correction is the dominant
two-electron contribution to the total fns effect. It is suppressed by the factor of 1/Z with respect to the leading one(0)
electron fns contribution δgN . The one-photon exchange fns
correction can be obtained as a difference of the one-photon
exchange contributions to the g-factor evaluated with the extended nuclear charge distribution and with the point nucleus,
(1)
(1)
(1)
δgN = δgext − δgpnt .
(15)
4
(0)
TABLE II: One-electron QED fns corrections to the bound-electron g factor, expressed in terms of GNQED defined by Eq. (13). The abbreviations are as follows: ”NSE” denotes the self-energy contribution, ”NUe,el” denotes the Uehling electric-loop vacuum-polarization correction,
”NWK,el” stands for the Wichmann-Kroll electric-loop vacuum-polarization correction, and ”NVP,ml” denotes the magnetic-loop vacuumpolarization contribution.
Z
6
8
10
12
14
20
25
30
35
40
45
50
55
60
NSE
NUe,el
−0.54 (20)
−0.77 (10)
−0.94 (4)
−1.14 (4)
−1.32 (4)
−1.86 (4)
−2.36 (4)
−2.82 (4)
−3.27 (2)
−3.75 (2)
−4.23 (1)
−4.73 (1)
−5.25 (1)
−5.79 (2)
0.179
0.256
0.337
0.430
0.530
0.863
1.185
1.543
1.933
2.376
2.837
3.348
3.902
4.515
NWK,el
NVP,ml
Total, 2s
Total, 1s
−0.011
−0.019
−0.028
−0.040
−0.053
−0.098
−0.143
−0.191
−0.240
−0.295
−0.345
−0.398
−0.450
−0.502 (1)
−0.010 (1)
−0.010 (1)
−0.013 (1)
−0.017 (2)
−0.018 (2)
−0.025 (4)
−0.030 (4)
−0.035 (6)
−0.039 (8)
−0.044 (8)
−0.047 (10)
−0.050 (12)
−0.053 (12)
−0.055 (14)
−0.38 (20)
−0.55 (10)
−0.65 (4)
−0.77 (4)
−0.86 (4)
−1.12 (4)
−1.35 (4)
−1.50 (4)
−1.62 (4)
−1.71 (2)
−1.79 (2)
−1.83 (1)
−1.85 (1)
−1.83 (2)
−0.60 (1)
−0.70 (1)
−0.807 (9)
−0.905 (8)
−0.996 (5)
−1.237 (3)
−1.404 (2)
−1.542 (2)
−1.655 (1)
−1.733 (1)
−1.793 (1)
−1.821 (1)
−1.819 (1)
−1.780 (1)
The one-photon exchange correction to the g-factor of the
ground and valence-excited states of Li-like ions is given
by [5]
h
XX
(1)
δg (1) = 2
(−1)P hP v P c|I(∆P c c )|δV v ci
µc
P
i
(1)
+ hP v P c|I(∆P c c )|v δV ci
X
−
hv|Vg |vi − hc|Vg |ci hcv|I ′ (∆vc )|vci , (16)
µc
where v and c denote the valence and the core electron states,
respectively, µc is the momentum projection of the core electron, P is the permutation operator, (P vP c) = (vc) or (cv),
(−1)P is the sign of the permutation, ∆ab = εa − εb , I(ω)
is the relativistic operator of the electron-electron interaction
defined below, and I ′ (ω0 ) = dI(ω)/(dω) at ω = ω0 . Further
(1)
notations used in Eq. (16) are as follows: δV a stands for the
first-order perturbation of the wave function a by the potential
Vg ,
(1)
|δV ai
=
εaX
6=εn
n
|ni hn|Vg |ai
,
εa − εn
I(ω, r1 , r2 ) = α (1 − α1 · α2 )
exp [i|ω|r12 ]
,
r12
(17)
(18)
where α is the vector of Dirac matrices in the standard representation. The above form of the potential Vg (r) assumes that
the momentum projection of the valence state v in Eq. (16) is
fixed as µv = 1/2.
The relativistic electron-electron interaction operator I(ω)
(19)
where r12 = |r1 − r2 | is the distance between the two electrons and ω is the frequency of the photon exchanged between
them.
The calculation of the one-photon exchange contribution
with the extended and the point nuclear models was reported
in Ref. [5]. In the present work, we redo these calculations
with an enhanced precision, which is necessary for an accurate identification of the fns effect. The one-photon exchange
(1)
fns correction δgN can be parameterized as
(1)
(0)
δgN = δgN
1 (1)
H (Zα, R) ,
Z
(20)
(0)
where δgN is the one-electron nuclear-size correction introduced earlier, and H (1) is a slowly varying function. The Zα
expansion of H (1) reads
H (1) = H (1,0) + (Zα)2 H (1,2+) ,
and Vg is the effective g-factor potential (see, e.g., Eq. (14) of
Ref. [19]),
Vg (r) = 2 m [r × α]z ,
in the Feynman gauge reads
(21)
where H (1,0) is the leading nonrelativistic contribution and
H (1,2+) is the higher-order remainder.
The nuclear-size correction is evaluated in this work as the
difference of Eq. (16) calculated with the extended vs. pointlike nuclear models. The numerical evaluation of Eq. (16)
with the extended nucleus is performed by using the DKB
method [16]. For the point nucleus, we use the analytical expressions for the reference-state wave functions and for the
diagonal (in κ) g-factor perturbed wave function [20], and the
standard implementation of the B-splines method [21] for the
non-diagonal in κ part of the perturbed wave function. In order to avoid loss of numerical accuracy in the low-Z region,
5
-2.84
-2.86
(1,0+)
-2.88
H
TABLE III: The one-photon exchange fns correction to the boundelectron g factor of the ground state of Li-like ions, in terms of the
function H (1) defined by Eq. (20). The column (R, c) contains results obtained with the actual values of the nuclear charge radii R
and the speed of light c. The column (4R, c) presents results obtained with the nuclear charge radii multiplied by a factor of 4. The
column (40R, 10c) contains results obtained with the nuclear charge
radii multiplied by a factor of 40 and the speed of light multiplied by
10.
-2.90
-2.92
(R,c)
(4 R,c)
(40 R,10 c)
-2.94
Z
6
8
10
12
14
20
25
30
35
40
45
50
55
60
(R, c)
−2.8529
−2.8538
−2.8550
−2.8566
−2.8584
−2.8654
−2.8731
−2.8824
−2.8933
−2.9057
−2.9194
−2.9346
−2.9510
−2.9686
(4R, c)
−2.8529
−2.8539
−2.8552
−2.8568
−2.8586
−2.8658
−2.8735
−2.8828
−2.8936
−2.9057
−2.9191
−2.9336
−2.9491
−2.9655
(40R, 10c)
−2.8527
−2.8533
−2.8539
−2.8545
−2.8550
−2.8569
−2.8585
−2.8601
−2.8616
−2.8629
−2.8642
−2.8653
−2.8663
−2.8670
we employ the DKB and the B-splines methods implemented
in the quadruple arithmetics.
The accuracy of the obtained numerical results is checked
as follows. We observe that the leading term of the Zα expansion of Eq. (21), H (1,0) , should not depend on the nuclear
charge radius R. It also cannot depend on the speed of light
c. All dependence of H (1,0+) on R and c comes only through
the relativistic effects, which are small corrections in the lowZ region. Therefore, numerical calculations of H (1,0+) performed with different choices of R and c should have the same
low-Z limit.
The numerical results for the nuclear-size correction to the
one-photon exchange are presented in Table III and shown
graphically on Fig. 1. We observe that the results obtained
with different values of R and c are in very good agreement
for low Z. This agreement also indicates that the results for
H (1) are practically independent of the nuclear model.
The results obtained with enlarged speed of light show very
weak Z dependence, which might have been anticipated since
the Z dependence of H (1) comes through the relativistic corrections only. These results can be easily extrapolated to
Z → 0, yielding
H (1,0) = −2.8512 (10) .
(22)
On the basis of this result, we conclude that the following
weighted difference of the 2s and 1s g-factors cancels most
of the fns contribution of order 1/Z for light ions,
2851 1
(1)
δΞ1 g = δg (2s) − Ξ0 −
g (0) (1s) .
(23)
1000 Z
-2.96
-2.98
0
10
20
30
40
50
60
Z
FIG. 1: (Color online) The one-photon exchange fns correction to
the bound-electron g factor of the ground state of Li-like ions, in
terms of the function H (1) defined by Eq. (20). Numerical results
for the actual values the nuclear charge radii and the speed of light
(R, c) (filled dots, red) are compared with the results obtained with
with the nuclear charge radii multiplied by a factor of 4 (4R, c) (filled
stars, blue) and with the results obtained with the nuclear charge radii
multiplied by a factor of 40 and the speed of light multiplied by 10
(40R, 10c) (open dots, green).
D. Two and more photon exchange fns correction
The fns correction with two and more photon exchanges
between the electrons is suppressed by the factor of 1/Z 2 with
respect to the leading fns contributions. A parametrization of
this term can be given as
(2+)
δgN
(0)
= δgN
1
H (2+) (Zα, R) ,
Z2
(24)
(0)
where δgN is the one-electron nuclear-size correction defined
in Eq. (3), and H (2+) is a slowly varying function of its arguments.
In order to compute the fns correction, we need to calculate
the two and more photon exchange correction for the extended
and the point nucleus and take the difference,
(2+)
δgN
(2+)
(2+)
= δgext − δgpnt .
(2+)
(2+)
(25)
In this work, we calculate δgext and δgpnt within the Breit
approximation. The whole calculation is performed in three
steps. In the first step, we solve the no-pair Dirac-CoulombBreit Hamiltonian by the Configuration-Interaction DiracFock-Sturm (CI-DFS) method [22]. In the second step, we
subtract the leading-order terms of orders 1/Z 0 and 1/Z 1 ,
thus identifying the contribution of order 1/Z 2 and higher.
The subtraction terms of order 1/Z 0 and 1/Z 1 were calculated separately by perturbation theory. In the third step, we
repeat the calculation for the extended and the point nuclear
models and, by taking the difference, obtain the fns correction.
The fns effect is very small in the low-Z region, which
makes it very difficult to obtain reliable predictions for this
6
(R, c)
(4R, c)
1.059 (20)
1.073 (20)
1.110 (20)
1.149 (20)
1.195 (20)
1.249 (20)
1.312 (20)
1.466 (20)
1.560 (20)
1.672 (20)
1.102 (20)
1.157 (20)
1.198 (20)
1.255 (20)
1.321 (20)
1.481 (20)
1.579 (20)
1.690 (20)
(40R, 10c)
1.081 (20)
1.075 (20)
1.075 (20)
1.074 (20)
1.074 (20)
1.073 (20)
1.072 (20)
1.068 (20)
1.067 (20)
1.064 (20)
1.9
1.8
(R,c)
(R*4,c)
(R*40,c*10)
1.7
1.6
1.5
(2+)
Z
10
14
20
25
30
35
40
50
55
60
where g(2s) is the g factor of the ground state of the Li-like
ion, g(1s) is the g factor of the ground state of the H-like
ion, and the weight parameter Ξ is defined by Eq. (2). Basing on the analysis of the preceding Section, we claim that
in the Ξ-weighted difference δΞ g, the nonrelativistic fns corrections to order 1/Z 0 , 1/Z 1 , and 1/Z 2 and, in addition, the
H
TABLE IV: The two and more photon exchange fns correction to the
bound-electron g factor of the ground state of Li-like ions, in terms
of the function H (2+) defined by Eq. (24). Notations are the same as
in Table III.
1.4
1.3
1.2
1.1
1.0
0.9
correction. In order to be able to monitor the numerical accuracy, we performed three sets of calculations. The first
set (R, c) was obtained with the actual values of the nuclear
charge radii R and the speed of light c; the second set (4R, c)
was obtained with the nuclear charge radii multiplied by a factor of 4; the third set (40R, 10c) was obtained with the nuclear
charge radii multiplied by a factor of 40 and the speed of light
multiplied by 10. The obtained results are listed in Table IV
and presented in Fig. 2.
Similarly to the one-photon exchange fns correction, we assume that the low-Z limit of H (2+) , denoted as H (2,0) , does
not depend either on R or on c. By extrapolating our numerical results in Table IV to Z → 0, we obtain the nonrelativistic
value of the 1/Z 2 correction as
H
(2,0)
= 1.070 (25) .
(26)
Based on this result, we conclude that for light ions, the following weighted difference of the 2s and 1s g-factors cancels
most of the 1/Z 2 fns contribution:
107 1
(2+)
(2s) − Ξ0
δΞ2 g = δg
g (0) (1s) .
(27)
100 Z 2
III.
THE WEIGHTED DIFFERENCE OF THE 2s AND 1s
g FACTORS
Combining the results obtained in the previous section, we
introduce the total Ξ-weighted difference as follows
δΞ g = g(2s) − Ξ g(1s) ,
(28)
0.8
0
10
20
30
40
50
60
Z
FIG. 2: (Color online) The two and more photon exchange fns correction to the bound-electron g factor of the ground state of Li-like
ions, in terms of the function H (2+) defined by Eq. (24). Notations
are the same as in Fig. 1.
relativistic contribution to order (Zα)2 /Z 0 are cancelled. A
small remaining fns correction to δΞ g is calculated numerically. The definition of δΞ g is based on the Zα expansion of
the fns corrections. Because of this, it is applicable for lowand medium-Z ions. For heavy systems, the Zα expansion
is no longer useful. In this case, the cancellation of the fns
effect in the weighted difference is still possible but should be
achieved differently [5, 6].
In Table V we present the individual fns contributions to
the g-factor of the ground state of Li-like ions g(2s), H-like
ions g(1s) and for the weighted difference δΞ g. We observe
that the uncertainty of the fns corrections for g(2s) and g(1s)
is dominated by the nuclear-model and nuclear-radii errors,
which means they cannot be significantly improved. On the
contrary, the fns effect for δΞ g is much smaller, and its uncertainty is mainly numerical, meaning that it can be improved
further.
7
-9
-9
10
-10
10
-11
10
-12
10
10
10
dg
-9
10
10
-13
10
-10
10
-10
-11
10
-12
10
-11
-12
-13
10
-13
10
g(2s)
-14
-14
10
10
δΞg
-15
10
-15
10
-15
10
0
10
20
30
40
50
60
δΩg
-14
10
10
0
Z
10
20
30
40
50
60
0
10
Z
20
30
40
50
60
Z
FIG. 3: (Color online) Comparison of the error δg = (∂g/∂α) δα due to the uncertainty of the fine-structure constant δα/α = 3.2 × 10−10
(solid line, green) and the error due to the finite nuclear size effect (dashed-dot line, red), for the g-factor of the ground state of Li-like ions
g(2s) (left panel); for the weighted difference δΞ g(Z) (middle panel); and for the weighted difference δΩ g = δΞ g(Z) − δΞ g([Z/2]) (right
panel).
TABLE V: The fns corrections to the bound-electron g factor of the ground state
of Li-like and H-like ions and their weighted difference, multiplied by a factor
of 106 . The numbers in the parentheses denote the uncertainty in the last figure. When three uncertainties are specified, the first one is the numerical error,
the second one the model-dependence error, and the third one the uncertainty induced by the error of the nuclear charge radius. In the case only one uncertainty
is specified, it is the numerical error (whereas the other errors are significantly
smaller and are not indicated).
Z Term
δgN (2s)
Ξi /Z i δgN (1s)
δgN (2s) − Ξi /Z i δgN (1s)
6 1/Z 0
α/Z 0
1/Z 1
1/Z 2+
Total
0.000 050 99 (0)(1)(9)
−0.000 000 05 (2)
−0.000 024 24 (0)(0)(4)
0.000 001 52 (4)
0.000 028 2 (0)(0)(1)
0.000 050 99 (0)(1)(9)
−0.000 000 071 (2)
−0.000 024 23 (0)(0)(4)
0.000 001 515
0.000 0282 (0)(0)(1)
0.
0.000 000 03 (2)
−0.000 000 016 (1)(0)(0)
0.000 000 00 (4)
0.000 000 01 (4)(0)(0)
8 1/Z 0
α/Z 0
1/Z 1
1/Z 2+
Total
0.000 194 7 (0)(0)(7)
−0.000 000 25 (3)
−0.000 069 5 (0)(0)(3)
0.000 003 26 (8)
0.000 128 3 (1)(0)(8)
0.000 194 7 (0)(0)(7)
−0.000 000 317 (5)
−0.000 069 4 (0)(0)(3)
0.000 003 256
0.000 128 3 (0)(0)(8)
0.
0.000 000 07 (3)
−0.000 000 068 (1)(0)(0)
0.000 000 00 (8)
0.000 000 00 (8)(0)(0)
10 1/Z 0
α/Z 0
1/Z 1
1/Z 2+
Total
0.000 598 3 (0)(1)(8)
−0.000 000 90 (8)
−0.000 170 8 (0)(0)(2)
0.000 006 4 (1)
0.000 433 0 (2)(1)(9)
0.000 598 3 (0)(1)(8)
−0.000 001 12 (1)
−0.000 170 6 (0)(0)(2)
0.000 006 40
0.000 433 0 (0)(1)(9)
−0.000 000 002
0.000 000 22 (8)
−0.000 000 241 (1)(0)(0)
0.000 000 0 (1)
0.000 000 0 (2)(0)(0)
12 1/Z 0
α/Z 0
1/Z 1
1/Z 2+
Total
0.001 307 (0)(0)(1)
−0.000 002 3 (2)
−0.000 311 1 (0)(1)(3)
0.000 009 7 (2)
0.001 003 (0)(0)(1)
0.001 307 (0)(0)(1)
−0.000 002 74 (2)
−0.000 310 5 (0)(1)(3)
0.000 009 71
0.001 003 (0)(0)(1)
−0.000 000 007
0.000 000 4 (2)
−0.000 000 604 (1)(0)(1)
0.000 000 0 (2)
−0.000 000 2 (3)(0)(0)
14 1/Z 0
α/Z 0
1/Z 1
1/Z 2+
Total
0.002 580 (0)(1)(4)
−0.000 005 1 (3)
−0.000 5267 (0)(2)(8)
0.000 014 1 (3)
0.002 062 (0)(1)(4)
0.002 580 (0)(1)(4)
−0.000 005 96 (3)
−0.000 525 3 (0)(2)(8)
0.000 014 1
0.002 062 (0)(1)(4)
−0.000 000 026 (0)(1)(0)
0.000 000 8 (3)
−0.000 001 353 (1)(0)(2)
0.000 000 0 (3)
−0.000 000 6 (4)(0)(0)
8
20 1/Z 0
α/Z 0
1/Z 1
1/Z 2+
Total
0.014 41 (0)(1)(2)
−0.000 038 (2)
−0.002 064 (0)(1)(2)
0.000 040 0 (7)
0.012 34 (0)(1)(2)
0.014 41 (0)(1)(2)
−0.000 041 4 (1)
−0.002 054 (0)(1)(2)
0.000 038 5
0.012 35 (0)(1)(2)
−0.000 000 554 (0)(7)(1)
0.000 004 (2)
−0.000 010 31 (0)(0)(1)
0.000 001 4 (7)
−0.000 006 (2)(0)(0)
25 1/Z 0
α/Z 0
1/Z 1
1/Z 2+
Total
0.043 36 (0)(3)(5)
−0.000 136 (5)
−0.004 983 (0)(3)(6)
0.000 080 (1)
0.038 32 (1)(3)(5)
0.043 36 (0)(3)(5)
−0.000 141 4 (2)
−0.004 945 (0)(3)(6)
0.000 074
0.038 35 (0)(3)(5)
−0.000 003 90 (0)(4)(1)
0.000 005 (5)
−0.000 037 92 (0)(2)(4)
0.000 006 (1)
−0.000 031 (5)(0)(0)
30 1/Z 0
α/Z 0
1/Z 1
1/Z 2+
Total
0.111 34 (0)(8)(8)
−0.000 39 (1)
−0.010 697 (0)(8)(8)
0.000 148 (2)
0.100 40 (1)(8)(8)
0.111 36 (0)(8)(8)
−0.000 398 9 (5)
−0.010 583 (0)(8)(8)
0.000 132
0.100 51 (0)(8)(8)
−0.000 020 3 (0)(1)(0)
0.000 01 (1)
−0.000 114 72 (0)(9)(9)
0.000 016 (2)
−0.000 11 (1)(0)(0)
35 1/Z 0
α/Z 0
1/Z 1
1/Z 2+
Total
0.258 8 (0)(2)(3)
−0.000 97 (2)
−0.021 40 (0)(2)(2)
0.000 265 (4)
0.236 7 (0)(2)(3)
0.258 9 (0)(2)(3)
−0.000 995 4 (6)
−0.021 09 (0)(2)(2)
0.000 226
0.237 1 (0)(2)(3)
−0.000 086 4 (0)(5)(1)
0.000 02 (2)
−0.000 305 7 (0)(3)(3)
0.000 039 (4)
−0.000 33 (2)(0)(0)
40 1/Z 0
α/Z 0
1/Z 1
1/Z 2+
Total
0.527 6 (0)(5)(2)
−0.002 10 (3)
−0.038 33 (0)(4)(2)
0.000 436 (7)
0.487 6 (0)(5)(2)
0.527 9 (0)(5)(2)
−0.002 125 (1)
−0.037 63 (0)(4)(2)
0.000 353
0.488 5 (0)(5)(2)
−0.000 298 (0)(1)(0)
0.000 03 (3)
−0.000 699 6 (0)(8)(3)
0.000 083 (7)
−0.000 89 (3)(0)(0)
45 1/Z 0
α/Z 0
1/Z 1
1/Z 2+
Total
1.076 (0)(1)(1)
−0.004 47 (3)
−0.069 81 (0)(8)(7)
0.000 74 (1)
1.003 (0)(1)(1)
1.077 (0)(1)(1)
−0.004 486 (3)
−0.068 24 (0)(8)(7)
0.000 569
1.005 (0)(1)(1)
−0.000 982 (0)(3)(1)
0.000 02 (4)
−0.001 574 (0)(2)(2)
0.000 17 (1)
−0.002 37 (4)(0)(0)
50 1/Z 0
α/Z 0
1/Z 1
1/Z 2+
Total
2.050 (0)(3)(2)
−0.008 73 (5)
−0.120 3 (0)(2)(1)
0.001 21 (2)
1.922 (0)(3)(2)
2.053 (0)(3)(2)
−0.008 684 (5)
−0.117 1 (0)(1)(1)
0.000 878
1.928 (0)(3)(2)
−0.002 885 (0)(7)(3)
−0.000 05 (5)
−0.003 262 (0)(5)(3)
0.000 34 (2)
−0.005 86 (5)(1)(0)
55 1/Z 0
α/Z 0
1/Z 1
1/Z 2+
Total
3.788 (0)(5)(7)
−0.016 29 (9)
−0.203 2 (0)(3)(4)
0.001 98 (3)
3.570 (0)(5)(7)
3.796 (0)(5)(7)
−0.016 037 (9)
−0.196 8 (0)(3)(3)
0.001 342
3.584 (0)(5)(7)
−0.007 95 (0)(1)(2)
−0.000 26 (9)
−0.006 47 (0)(1)(1)
0.000 63 (3)
−0.014 05 (9)(2)(2)
60 1/Z 0
α/Z 0
1/Z 1
1/Z 2+
Total
6.74 (0)(1)(1)
−0.028 7 (2)
−0.333 6 (0)(5)(3)
0.003 17 (4)
6.38 (0)(1)(1)
6.76 (0)(1)(1)
−0.027 96 (2)
−0.321 4 (0)(5)(3)
0.002 010
6.42 (0)(1)(1)
−0.020 51 (0)(2)(2)
−0.000 8 (2)
−0.012 24 (0)(2)(1)
0.001 16 (4)
−0.032 4 (2)(0)(0)
We would like now to address the question whether the
weighted difference δΞ g might be useful for the determination of the fine-structure constant α. The leading dependence
of δΞ g on α is given by the expansion
1
α
2
− Ξ + (1 − Ξ) + . . . ,
δΞ g = 2 (1 − Ξ) − (Zα)2
3
4
π
(29)
where the second term in the right-hand-side stems from the
9
binding corrections, whereas the third term is due to the oneloop free-electron QED effect. In the above equation, we keep
Ξ fixed, ignoring its dependence on α, since it does not contribute to the sensitivity of δΞ g on α (the same value of Ξ
should be used when comparing the experimental and theoretical values of δΞ g). By varying α in Eq. (29) within its current
error bars of δα/α = 3.2 × 10−10 [4], the corresponding error
of δΞ g can be obtained.
In Fig. 3 we compare the uncertainty due to α and the
uncertainty due to the nuclear model and radius, keeping in
mind that the latter defines the ultimate limit of the accuracy
of theoretical calculations. The left panel of Fig. 3 shows
this comparison for the g-factor of the ground state of Li-like
ions g(2s), whereas the middle panel gives the same comparison for the Ξ-weighted difference δΞ g. The dip of the
α-sensitivity curve around Z = 16 is caused by the fact that
the dependence of the binding and the free-QED effects on
α in Eq. (29) (second and third terms) have different signs,
and thus cancel each other in this Z region. From Fig. 3 we
can conclude that up to Z ≈ 45, the weighted difference δΞ g
yields possibilities for an improved determination of α.
The determination of α from δΞ g has two drawbacks. The
first one is the cancellation of α dependence of δΞ g around
Z = 16, leading to a loss of sensitivity to α in this Z region. The second one is that δΞ g contains the same free-QED
part which is used for the determination of α from the freeelectron g factor, which means that these two determinations
cannot be regarded as fully independent. Both drawbacks can
be avoided by introducing another difference,
δΩ g = δΞ g(Z) − δΞ g([Z/2]) ,
(30)
with δΞ g(Z) being the weighted difference (28) for the nuclear charge Z, and δΞ g([Z/2]) is the corresponding difference for the nuclear charge [Z/2], where [. . .] stands for the
upper or the lower integer part. In the difference δΩ g, most
free-QED contributions vanish. So, by a small sacrifice of
the sensitivity of the binding effects to α, we removed the dip
around Z = 16 and made the theory of the weighted difference (almost) independent on the theory of the free-electron
g-factor.
The right panel of Fig 3 presents the comparison of the
uncertainty due to α with the error of the fns effect for the
weighted difference δΩ g. One finds a smooth dependence of
the sensitivity to α on Z, without any dip in the region around
Z = 16. We observe that in the region Z = 10 − 20, the
weighted difference δΩ g offers better possibilities for determining α than δΞ g.
Employing the difference δΩ g can be also advantageous
from the experimental point of view. It can be rewritten as
δΩ g = g(2s, Z) − g(2s, Z2 )
− Ξ(Z) [g(1s, Z) − g(1s, Z2 )]
− g(1s, Z2 ) [Ξ(Z) − Ξ(Z2 )] ,
(31)
with Z2 = [Z/2]. We thus observe that δΩ g can be effectively
determined in an experiment by measuring two equal-weight
g-factor differences (namely, the ones in the first and second
rows of the above equation) and g(1s, Z2 ). The equal-weight
differences may be measured with largely suppressed systematic errors and thus can be determined in near-future experiments much more accurately than the g-factors of individual
ions. The last term in Eq. (31) is suppressed by a small factor
of [Ξ(Z) − Ξ(Z2 )] ≈ 0.02 − 0.04 in the region of interest.
Therefore, the experimental error of δΩ g can be significantly
improved as compared to that of the absolute g-factors.
Let us now turn to the experimental consequences of the
present calculations. So far, the only element for which the
weighted difference δΞ g has been measured is silicon. In
Table VI we collect the individual theoretical contributions
to δΞ g(29 Si). Theoretical results for various effects were
taken from the literature, Refs. [9, 23–28]. The total theoretical value is compared to the experimental result [2, 29, 30].
The errors of the Dirac value and of the one-loop free QED
(∼ α(Zα)0 ) result specified in the table are due to the uncertainty of the current value of α−1 = 137.035 999 074 (44)
[4]. The uncertainty of the fns effect specified in the table is
6 × 10−13 , which is already smaller than the uncertainty of
the Dirac value due to α. The fns uncertainty is of purely numerical origin, i.e. it does not influenced by the errors due to
the rms charge radius and the nuclear charge distribution, and
thus it can be further improved in future calculations.
Table VI illustrates another advantage of the Ξ-weighted
difference: the contributions of one-electron binding QED effects to δΞ g are much smaller than those to g(2s). This is
explained by the fact that these effects largely originate from
short distances, similarly to the fns effect, and thus are significantly canceled in the difference. In particular, the uncertainty of δΞ g(Si) due to three-loop binding QED effects is on
the 10−12 level, implying that these effects do not need to be
known to a high degree of accuracy for the determination of
α.
Table VI shows that the present experimental and theoretical precision of δΞ g(Si) is on the level of few parts in 10−9 ,
which is significantly worse than the precision achieved for
other systems (in particular, H-like carbon, where the present
experimental and theoretical uncertainties are, correspondingly, 6 × 10−11 and 6 × 10−12 [1]). This underperformance
is, however, more due to a lack of motivation than due to principal obstacles.
On the experimental side, the same precision as for Hlike carbon can be also obtained for δΞ g(C), with an existing
ion trap [3]. Further experimental advance is anticipated that
could bring one or two orders of magnitude of improvement
[3]. On the theoretical side, the modern nonrelativistic quantum electrodynamics (NRQED) approach (see, e.g., [31]) can
apparently provide a theoretical result for Li-like carbon with
the same accuracy as obtained for its H-like counterpart [32].
Moreover, further theoretical advance is possible: the twoloop QED corrections of order α2 (Zα)5 and the three-loop
QED corrections of order α3 (Zα)4 can be calculated, both
for H-like and Li-like ions [32].
As we are presently interested in light ions, the best way
for the advancement of theory would be a combination of
two complementary methods. The first one is the NRQED
method (used, e.g., in [33]) that accounts for the nonrelativistic electron-electron interactions to all orders in 1/Z, but ex-
10
pands the QED and relativistic effects in powers of α and Zα.
The second approach (used, e.g., in [24, 26–28]) accounts for
the relativistic effects to all orders in Zα but employs perturbation expansions in α (QED effects) and in 1/Z (electronelectron interaction). Matching the coefficients of the Zα and
1/Z expansions from the two methods allows one to combine
them together, as it was done for energy levels in Ref. [34]. As
a result of this procedure, only higher-order corrections in Zα
will be expanded in 1/Z and only higher-order corrections in
1/Z will be expanded in Zα. This approach should allow one
to advance theory to the level required for a determination of
α.
The principal limitation for the theory is set by the nontrivial nuclear structural effects, such as the nuclear deformation, nuclear polarization, etc. For light ions, the leading nuclear effects are described by effective operators proportional
to the Dirac delta function δ(r). Such effects are canceled
in the weighted difference δΞ g. We estimate that the uncertainty due to the remaining nuclear effects in δΞ g should be
of the same order as the nuclear-model dependence error of
the fns effect. From the breakdown in Table V we deduce
that for silicon, this error is by about two orders of magnitude smaller than the uncertainty due to α. We thus conclude
that the nuclear effects do not represent any obstacles for the
determination of α from δΞ g and δΩ g.
IV.
CONCLUSION
In this work we investigated specific weighted differences
of the g-factors of H- and Li-like ions of the same element.
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Acknowledgments
V.A.Y. and Z.H. acknowledge helpful conversations with
Sven Sturm. V.A.Y. acknowledges support by the Ministry of
Education and Science of the Russian Federation (program for
organizing and carrying out scientific investigations) and by
RFBR (grant No. 16-02-00538). E.B. acknowledges support
from G-RISC, project No. P-2014a-9.
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11
TABLE VI: Individual contributions to the weighted difference δΞ g for 29 Si, M/m = 52806.93396, Ξ = 0.101136233077060.
Contribution
Dirac
1-loop QED
Order
α(Zα)0
α(Zα)2
α(Zα)4
α(Zα)5+
2-loop QED
α2 (Zα)0
α2 (Zα)2
α2 (Zα)4
α2 (Zα)5+
≥ 3-loop QED
α3+ (Zα)0
α3+ (Zα)2
α3+ (Zα)4+
Recoil
m/M (Zα)2+
1-photon exchange
(1/Z)(Zα)2+
2-photon exchange
(1/Z 2 )(Zα)2+
≥ 3-photon exchange (1/Z 3+ )(Zα)2+
2-electron QED
(α/Z)(Zα)2+
2-electron Recoil
(m/M )(1/Z)(Zα)2+
Finite nuclear size
Total theory
Experiment [2, 29]
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Value
1.796 687 854 216 5 (7)
0.002 087 898 255 0 (7)
0.000 000 601 506 0
0.000 000 014 797 0
0.000 000 015 48 (52)
−0.000 003 186 116 6
−0.000 000 000 917 9
−0.000 000 000 084 4
0.000 000 000 00 (13)
0.000 000 026 514 9 (1)
0.000 000 000 007 6
0.000 000 000 000 0 (11)
0.000 000 029 4 (10)
0.000 321 590 803 3
−0.000 006 876 0 (5)
0.000 000 093 0 (60)
−0.000 000 236 0 (50)
−0.000 000 011 6 (7)
−0.000 000 000 000 6 (4)
1.799 087 813 2 (79)
1.799 087 812 5 (21)
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